Unimodular Fourier multipliers on homogeneous Besov spaces

In this paper, we study the Fourier multiplier operator eiμ(D) on homogeneous Besov spaces B˙p,qs. If μ is a homogeneous function with positive degree whose Hessian matrix is non-degenerate at some point, we find the necessary conditions of pi,qi,si(i=1,2) for the boundedness of eiμ(D) from B˙p1,q1s1 to B˙p2,q2s2. Moreover, under a global non-degenerate assumptions on the Hessian matrix of μ(ξ), we obtain the sufficient and necessary conditions for the boundedness of eiμ(D) between B˙p1,q1s1 and B˙p2,q2s2. More precisely, we obtain the estimates of the operator norm ‖eiμ(D)‖B˙p1,q1s1→B˙p2,q2s2, and get the blowup rate near singularity points.